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Green's theorem

Problem 12.115 · easy

Use Green's theorem to evaluate \( \displaystyle \oint_C (x y)\,dx + (x^{2})\,dy \), where C is the boundary of the rectangle \( \displaystyle [0, 2] \times [0, 3] \), counterclockwise.

Green's theorem: ∮ P dx + Q dy = ∬ (∂Q/∂x − ∂P/∂y) dA.
Reviewed
\[ \frac{d}{d x} x^{2} - \frac{\partial}{\partial y} x y = x \]
∂Q/∂x − ∂P/∂y.✓ Proved
\[ \int\limits_{0}^{3}\int\limits_{0}^{2} x\, dx\, dy = 6 \]
Integrate over the rectangle.✓ Proved

Answer \( 6 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the line integral around the four edges, done directly, gives the same value

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies Green's theorem, computes the partial derivatives accurately, and sets up the double integral over the specified rectangular region with correct bounds.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies Green's theorem, computes the partial derivatives accurately, and sets up the double integral over the specified rectangular region with correct bounds.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies Green's theorem, computes the partial derivatives accurately, and evaluates the double integral over the specified rectangular region.
gpt-oss:20b: pass 2026-09-27

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/greens_theorem, checked 2026-09-27 with SymPy 1.14.0.